Question

which of these formulas gives the standard deviation? choose 1 answer: a) $s_x = \sqrt{\frac{(21 - 8)^2 + (28 - 8)^2 + \cdots + (70 - 8)^2}{47}}$ b) $s_x = \sqrt{\frac{(21 - 47)^2 + (28 - 47)^2 + \cdots + (70 - 47)^2}{7}}$ c) $s_x = \sqrt{\frac{(21 - 8) + (28 - 8) + \cdots + (70 - 8)}{47}}$ d) $s_x = \sqrt{\frac{(21 - 47) + (28 - 47) + \cdots + (70 - 47)}{7}}$ e) $s_x = \frac{21 + 28 + \cdots + 65 + 70}{8}$

Explanation:

Step1: Recall the sample standard deviation formula

The formula for the sample standard deviation \( s_x \) is \( s_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}} \), where \( \bar{x} \) is the sample mean, \( x_i \) are the data points, and \( n \) is the number of data points. First, we need to find the sample mean \( \bar{x} \) of the data set (21, 28, ..., 70). The mean \( \bar{x}=\frac{21 + 28+\cdots+70}{n} \). Let's assume the number of data points \( n \) (from the options, we can see denominators like 7, 47, 8. Let's check the mean: if we consider the data points, let's see the option B has denominator 7, and the mean is 47. Let's verify: the formula in B is \( s_x=\sqrt{\frac{(21 - 47)^2+(28 - 47)^2+\cdots+(70 - 47)^2}{7}} \). So first, the mean \( \bar{x} \) should be 47. Then, the sum of squared deviations from the mean, divided by \( n - 1 \) (but in sample standard deviation, it's \( n - 1 \), but let's check the options. Wait, maybe it's a population standard deviation? No, the formula for sample standard deviation is with \( n - 1 \), but let's check the options. Let's analyze each option:

• Option A: The deviation is from 8, not the mean. So incorrect.
• Option B: The deviation is from 47 (which is likely the mean), squared deviations, sum, divided by 7 (maybe \( n - 1 \) or \( n \), but let's check the formula structure. The standard deviation formula has squared deviations from the mean, sum, divided by (n - 1) for sample, or n for population, then square root. Here, the numerator is sum of \( (x_i-\bar{x})^2 \), denominator is 7, and square root. This matches the structure of standard deviation (either sample or population, but the key is squared deviations from mean).
• Option C: The numerator is sum of (x_i - 8), not squared, so incorrect (standard deviation involves squared deviations to eliminate sign issues).
• Option D: The numerator is sum of (x_i - 47), not squared, so incorrect.
• Option E: This is the formula for the mean, not standard deviation.

So the correct formula is the one with squared deviations from the mean (47) and then square root of the sum divided by the appropriate number (here 7).

Step2: Confirm the mean and formula structure

The mean \( \bar{x}=\frac{21 + 28+\cdots+70}{n} \). If we assume the number of data points is 8 (since option E has denominator 8), then \( \bar{x}=\frac{21 + 28+\cdots+70}{8} \). Let's check what \( \frac{21 + 28+\cdots+70}{8} \) is. Let's see, 21, 28, ..., 70. Let's assume there are 8 data points. Then the mean would be \( \frac{\text{sum}}{8} \). If the mean is 47, then sum is \( 47\times8 = 376 \). Let's check 21 + 28 + ... + 70. But regardless, the key is that standard deviation formula requires squared deviations from the mean, sum, divided by (n - 1) (or n) and square root. Option B has squared deviations from 47 (mean), sum, divided by 7 (which would be n - 1 if n = 8, since 8 - 1 = 7), and square root. So this matches the sample standard deviation formula.

Answer:

B. \( s_{x}=\sqrt{\frac{(21 - 47)^{2}+(28 - 47)^{2}+\cdots+(70 - 47)^{2}}{7}} \)